Will it become warmer if the volcano erupts?

Yes, it will.
It is even said that the temperature of the earth since its birth
has been basically determined by the content of carbonate gas in the air.

Budori:

If the Carbonado Island volcano erupts now, will it emit
carbonate gas much enough to change the climate?

Dr. Kûbô:

Yes, I have calculated it.
If it erupts, its gas will soon join the upper-level winds of the general circulation and will cover the whole earth.
It will prevent radiation of heat from the lower atmosphere and from the surface,
and I think that it will warm the whole globe by five degrees on the average.

--- MIYAZAWA Kenzi, 1932:
Gusukô Budori no Denki (A Biography of Gusukô Budori).
Translation of the quotation from Japanese is mine.

This piece of a juvenile novel
may as well be regarded as a science fiction
-- maybe one of the few science fictions that take the motif of global warming.
And it is also a humanistic story.
After this scene, Budori decides to sacrifice himself in a dangerous
mission of inducing an eruption.
As a teacher of soil science and geology, and later as a lonely merchant,
MIYAZAWA Kenzi [note 1]
(1896--1933) struggled to improve agriculture by means of science.
In the northeastern part of Japan near the margin of rice cultivation,
cool summers may even be fatal to rural economy.
Though he could not do it in the real world, climate modification toward
a warmer world seemed to him something good that science (plus human will)
would bring some day.

Even though Kenzi was a kind of genius,
he did not invent the theory of global warming due to carbon dioxide
(CO2, also known as carbonate gas) by himself.
As a scholar of literature investigated (ÔTSUKA,
1993), it is almost sure that Kenzi had read a book written by a Swedish
scientist Svante ARRHENIUS translated
to Japanese by ITINOHE Naozô.
(Though there are many editions, one example is
ARRHENIUS/ITINOHE 1914,
which is a translation of
ARRHENIUS/BAMBERGER 1907.)
ARRHENIUS (1859--1927),
best known as a chemist who developed
the theory of acids and bases, was also essential in the development of
the concept often called "greenhouse effect" (see, for example,
WEART, 2003, Chapter 1).
He also wrote books that popularized science.
One of them about the evolution of the universe included
a chapter of the history of the earth, where he attributed the climatic
changes to changing levels of CO2.
ITINOHE (1877--1920), also born in northeastern Japan
and educated as an astronomer,
was probably the first dedicated popularizer of natural science in Japan
(NAKAYAMA, 1980).
He struggled to issue the first science magazine in Japan.
Though rather amateur than professional, Kenzi was also interested in
astronomy.
It was not just incidental that he met the theory of
ARRHENIUS.

Now in the early 21st century, most of experts think that volcanic eruption
is likely to cool the climate rather than warm it.
It is because eruption brings much aerosols
--small particles floating in the air--.
Volcanic gas contains sulfur dioxide (SO2),
which becomes droplets of sulfuric acid (H2SO4)
or crystals of sulfates such as ammonium sulphate
((NH4)2SO4).
If the gas is injected to the stratosphere and aerosols are formed there,
they will stay there for a few years.
The effect that these aerosols reflect sunlight back to space overwhelms
their own greenhouse effect and the greenhouse effect of CO2
provided by the eruption together.
So, was Budori guided by a calculation which was to be proved false?

Perhaps beyond the intent of the author,
this story tells that
climate modification involves some global risks
(not just such local risks that can be taken by such
courageous persons as Budori).

But, an excuse may lie in the name of the volcano.
At least in a piece of fantasy, one can imagine a volcano that is very rich
in carbonates but lacks sulfates.

Another excuse is that it is not sure whether Budori was a member of
the human race as we are.
In an unfinished draft of the story,
it was explicit that the hero (not yet called Budori) was a
member of a race of monster rather than human.
(And in another version,
the way of attempted climate modification was
to remove black "pricks" from the sun!)
The change of the CO2 level caused by volcanic eruptions
lasts hundreds of years.
(Though exchanges of considerable amount of CO2 occur
between the atmosphere and the biosphere in a shorter time scale,
they are largely balanced.)
Thus, if the average life expectancy of Budori's race is
many hundreds of years, his action seems to be well-founded.

Note

Many people spell the personal name of MIYAZAWA
as "Kenji".
This is the transliteration in the HEPBURN system
[see another page of mine].
I suspect that he used Nippon-siki (Japanese system),
and temporarily guess that his preferred spelling was "Kenzi",
but it is also probable that it was "Kendi".